Optimal. Leaf size=281 \[ \frac {A (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (1+m;-\frac {5}{2},-\frac {5}{2};2+m;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (1+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {B (e x)^{2+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (2+m;-\frac {5}{2},-\frac {5}{2};3+m;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (2+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {857, 773, 138}
\begin {gather*} \frac {A (e x)^{m+1} \left (a+b x+c x^2\right )^{5/2} F_1\left (m+1;-\frac {5}{2},-\frac {5}{2};m+2;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (m+1) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}}+\frac {B (e x)^{m+2} \left (a+b x+c x^2\right )^{5/2} F_1\left (m+2;-\frac {5}{2},-\frac {5}{2};m+3;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 773
Rule 857
Rubi steps
\begin {align*} \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=A \int (e x)^m \left (a+b x+c x^2\right )^{5/2} \, dx+\frac {B \int (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} \, dx}{e}\\ &=\frac {\left (B \left (a+b x+c x^2\right )^{5/2}\right ) \text {Subst}\left (\int x^{1+m} \left (1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1+\frac {2 c x}{\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \, dx,x,e x\right )}{e^2 \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {\left (A \left (a+b x+c x^2\right )^{5/2}\right ) \text {Subst}\left (\int x^m \left (1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \left (1+\frac {2 c x}{\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{5/2} \, dx,x,e x\right )}{e \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}\\ &=\frac {A (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (1+m;-\frac {5}{2},-\frac {5}{2};2+m;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (1+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {B (e x)^{2+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (2+m;-\frac {5}{2},-\frac {5}{2};3+m;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (2+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(618\) vs. \(2(281)=562\).
time = 1.91, size = 618, normalized size = 2.20 \begin {gather*} \frac {x (e x)^m \sqrt {a+x (b+c x)} \left (a^2 A \left (720+1044 m+580 m^2+155 m^3+20 m^4+m^5\right ) F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(1+m) x \left (a (2 A b+a B) \left (360+342 m+119 m^2+18 m^3+m^4\right ) F_1\left (2+m;-\frac {1}{2},-\frac {1}{2};3+m;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(2+m) x \left (\left (2 a b B+A \left (b^2+2 a c\right )\right ) \left (120+74 m+15 m^2+m^3\right ) F_1\left (3+m;-\frac {1}{2},-\frac {1}{2};4+m;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(3+m) x \left (\left (b^2 B+2 A b c+2 a B c\right ) \left (30+11 m+m^2\right ) F_1\left (4+m;-\frac {1}{2},-\frac {1}{2};5+m;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+c (4+m) x \left ((2 b B+A c) (6+m) F_1\left (5+m;-\frac {1}{2},-\frac {1}{2};6+m;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+B c (5+m) x F_1\left (6+m;-\frac {1}{2},-\frac {1}{2};7+m;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )\right )\right )\right )}{(1+m) (2+m) (3+m) (4+m) (5+m) (6+m) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{m} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e\,x\right )}^m\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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